Sobolev Spaces, Embeddings, and Bounded Variation

Integration by parts characterizes derivatives through their action on smooth test functions. This characterization continues to make sense for functions that are not classically differentiable. Sobolev spaces require the resulting weak derivatives to be integrable, while bounded variation allows them to be measures.

We first study approximation by smooth functions and the role of the boundary. Trace and extension operators then let us pass between a domain and its surroundings. The embedding and compactness estimates explain what control of weak derivatives tells us about the functions themselves.

Definition 1. Let \(U \subset \mathbb R^n\) be an open subset. Let \(f \in L^1_{\rm loc} (U)\). \(g_i \in L^1_{\rm loc} (U)\) is the weak partial derivative of \(f\) with respect to \(x_i\) in \(U\) if

\[\begin{aligned} \int_U f \dpd{\varphi}{{x_i}} \dif x = - \int_U g_i \varphi \dif x ~~,~\text{for all } \varphi \in \mathscr C^1_c (U) ~. \end{aligned}\]

If two weak partial derivatives exist, their difference integrates to zero against every smooth compactly supported test function. Convolution of this difference with a mollifier is therefore zero on every smaller open set. Letting the mollification scale tend to zero gives equality almost everywhere. Thus the weak partial derivative is unique \(\lambda_n\)-a.e. \(x\). We can now define the following objects.

Definition 2. For \(p \in [1,\infty]\), \(W^{1,p} (U)\) is the Sobolev space if it contains all functions \(f \in L^p (U)\) and all weak partial derivatives \(\pd{f}{{x_i}}\) exist and each belong to \(L^p (U)\) for all \(i\). We say \(f \in W^{1,p}_{\rm loc} (U)\) if \(f \in W^{1,p} (V)\) for open sets \(V \subset_c U\) (i.e. \(\overline V\) is compact and \(\overline V \subset U\)). Lastly, \(f\) is called a Sobolev function if \(f \in W^{1,p}_{\rm loc} (U)\) for some \(p \in [1,\infty]\).

Definition 3. For \(f \in W^{1,p} (U)\), the norm is defined as

\[\begin{aligned} \|f\|_{W^{1,p} (U)} \triangleq \left( \int_U |f|^p + |\Dif f|^p \dif x \right)^{\frac1p} ~. \end{aligned}\]

This holds for any \(p \in [1,\infty)\). For \(p = \infty\),

\[\begin{aligned} \|f\|_{W^{1,\infty} (U)} \triangleq \operatorname{ess} \sup_U \left( |f| + |\Dif f| \right) ~. \end{aligned}\]

Definition 4. We say \(f_k \to f\) in \(W^{1,p}(U)\) if \(\|f_k - f\|_{W^{1,p} (U)} \to 0\). Similarly, we say \(f_k \to f\) in \(W^{1,p}_{\rm loc}(U)\) if \(\|f_k - f\|_{W^{1,p} (V)} \to 0\) for all \(V \subset_c U\).

Convolution with a smooth kernel averages a function over a small neighbourhood. This produces smooth approximations and, away from the boundary, commutes with weak differentiation. To do so, define, for \(\varepsilon > 0\), \(U_\varepsilon = \{x \in U : \mathop{\mathrm{\mathrm{dist}}}(x, \partial U) > \varepsilon\}\). Define

\[\begin{aligned} \eta(x) \triangleq \begin{cases} c \exp((|x|^2-1)^{-1}) & \text{ if } |x| < 1 ~, \\ 0 & \text{ otherwise} ~. \end{cases} \end{aligned}\]

Define the constant \(c\) such that \(\int_{\mathbb R^n} \eta(x) \dif x = 1\). Then, define

\[\begin{aligned} \eta_\varepsilon (x) \triangleq \varepsilon^{-n} \eta \left( \frac{x}{\varepsilon} \right) ~~, \varepsilon > 0 ~\text{and}~ x \in \mathbb R^n ~. \end{aligned}\]

This is called the mollifier. Finally, for any \(f \in L^1_{\rm loc} (U)\), define

\[\begin{aligned} f^\varepsilon \triangleq \eta_\varepsilon \ast f \iff f^\varepsilon (x) = \int_U \eta_\varepsilon (x-y) f(y) \dif y \quad \text{for } x \in U_\varepsilon ~. \end{aligned}\]

The following properties record smoothness, convergence, and compatibility with weak derivatives.

Theorem 5.

For each \(\varepsilon > 0\), \(f^\varepsilon \in \mathscr C^\infty (U_\varepsilon)\).
For \(f \in \mathscr C(U)\), \(f^\varepsilon \to f\) uniformly on compact subsets of \(U\).
\(f^\varepsilon \to f\) in \(L^p_{\rm loc} (U)\) if \(f \in L^p_{\rm loc} (U)\) for some \(p \in [1,\infty)\).
\(f^\varepsilon \to f\) \(\lambda_n\)-a.e. \(x\) (this convergence is understood in the pointwise sense).
If \(f \in W^{1,p}_{\rm loc} (U)\) for some \(p \in [1,\infty]\), then

\[\begin{aligned} \dpd{{f^{\varepsilon}}}{{x_i}} = \eta_\varepsilon \ast \dpd{f}{{x_i}} \end{aligned}\]

on \(U_\varepsilon\) for any \(i\).
If \(f \in W^{1,p}_{\rm loc} (U)\) for some \(p \in [1,\infty)\), then \(f^\varepsilon \to f\) in \(W^{1,p}_{\rm loc} (U)\).

Proof of Theorem 5. On a compact subset of \(U_\varepsilon\), differentiating the smooth kernel under the integral is justified by a common compact support and local integrability of \(f\). This proves smoothness. If \(f\) is continuous, then

\[|f^\varepsilon(x)-f(x)|\leq\sup_{|z|\leq\varepsilon}|f(x-z)-f(x)|,\]

which tends uniformly to zero on compact subsets. For \(f\in L^p_{\rm loc}\), first multiply by a cutoff that equals one on a neighbourhood of the compact set in question. Translation continuity in \(L^p(\mathbb R^n)\), for \(p<\infty\), and Minkowski’s inequality give

\[\|\eta_\varepsilon*f-f\|_p\leq\int\eta(z)\|f(\cdot-\varepsilon z)-f\|_p\,dz\longrightarrow0.\]

Translation continuity follows first for continuous functions of compact support by uniform continuity, and then for \(L^p\) functions by density. At a Lebesgue point \(x\), boundedness of the kernel gives

\[|f^\varepsilon(x)-f(x)|\leq C\fint_{B_\varepsilon(x)}|f(y)-f(x)|\,dy\longrightarrow0.\]

The Lebesgue differentiation theorem supplies almost every point. Finally, for \(x\in U_\varepsilon\), the function \(y\mapsto\eta_\varepsilon(x-y)\) is an admissible test function. The weak derivative identity gives

\[\partial_i f^\varepsilon(x)=-\int f(y)\partial_{y_i}\eta_\varepsilon(x-y)\,dy =\int\partial_i f(y)\eta_\varepsilon(x-y)\,dy.\]

Apply the \(L^p\) approximation already proved to \(f\) and each weak derivative to obtain the last assertion. The derivative identity also holds for \(p=\infty\), but norm convergence in \(W^{1,\infty}\) is not asserted.

Theorem 6. For any open \(U\) and \(1\leq p<\infty\), every \(f\in W^{1,p}(U)\) is a limit in the global \(W^{1,p}(U)\) norm of functions in \(C^\infty(U)\cap W^{1,p}(U)\).

Proof. Take a locally finite smooth partition of unity \((\psi_j)\) on \(U\), with compact supports contained in a locally finite family of relatively compact open sets \(V_j\). The weak product rule, obtained directly by testing with \(\psi_j\varphi\), is

\[D(\psi_jf)=\psi_jDf+fD\psi_j.\]

Since \(\psi_jf\) vanishes near the boundary of \(U\), its zero extension belongs to \(W^{1,p}(\mathbb R^n)\). For a prescribed \(\delta>0\), choose \(\varepsilon_j>0\) so small that \(g_j=\eta_{\varepsilon_j}*(\psi_jf)\) has support in \(V_j\) and

\[\|g_j-\psi_jf\|_{W^{1,p}(\mathbb R^n)}<\delta2^{-j}.\]

The locally finite sum \(g=\sum_jg_j\) is smooth on \(U\). Its difference from \(f=\sum_j\psi_jf\) is an absolutely convergent sum in \(W^{1,p}\), so the triangle inequality gives \(\|g-f\|_{W^{1,p}(U)}<\delta\). Taking \(\delta\downarrow0\) proves the claim. The choice of separate scales is needed because a single convolution scale cannot generally stay inside an arbitrary open set.

We can upgrade the statement of Theorem 6 via a kind of “topological regularity” condition, which some texts call the Lipschitz property of the topological boundary.

Definition 7. Let \(U \subset \mathbb R^n\) be an open set. The topological boundary \(\partial U\) is Lipschitz if, for all \(x \in \partial U\), there exists some \(r > 0\) and a Lipschitz map \(\gamma : \mathbb R^{n-1} \to \mathbb R\) such that

\[\begin{aligned} U \cap Q(x,r) = \{y : \gamma(y_1, \cdots, y_{n-1}) < y_n\} \cap Q(x,r) ~, \end{aligned}\]

for \(Q(x,r) \triangleq \{y : |y_i - x_i| < r ~\text{for all } i\}\). (This definition permits relabelling and rotating coordinate axes.)

A Lipschitz boundary chart of a domain U: the cube Q(x,r) about a boundary point x with its inward normal n(x), the smaller cube Q-tilde(x,r-tilde), and a ball of radius epsilon about the inward-shifted point y^epsilon that stays inside U. — Lipschitz boundary.

Rademacher’s theorem and the area formula imply that the graph has a tangent plane and an outward unit normal at \(\mathcal H^{n-1}\)-almost every boundary point. In coordinates where \(U\) lies above \(\gamma\), this normal is

\[\nu=\frac{(D\gamma,-1)}{\sqrt{1+|D\gamma|^2}}.\]

The Lipschitz bound also gives a uniform inward shift on each boundary chart, which is the geometric ingredient in the next proof.

Theorem 8. If \(U\) is bounded and open with Lipschitz boundary and \(1\leq p<\infty\), then every \(f\in W^{1,p}(U)\) has approximations \(f_k\in C^\infty(\overline U)\) converging in \(W^{1,p}(U)\). Here \(C^\infty(\overline U)\) means restriction of a smooth function on a neighbourhood of \(\overline U\).

Proof. Localize first to a boundary chart by a smooth cutoff whose support stays away from the edges of the chart. Write \(U=\{y_n>\gamma(y')\}\) there, let \(L\) be a Lipschitz constant for \(\gamma\), and fix \(\alpha>L+1\). For \(y\in\overline U\) in a smaller chart and \(|z|<\varepsilon\),

\[y_n+\alpha\varepsilon-z_n-\gamma(y'-z') \geq(\alpha-1-L)\varepsilon>0.\]

For small enough \(\varepsilon\), these points also stay in the larger chart. Thus the entire ball \(B_\varepsilon(y+\alpha\varepsilon e_n)\) lies inside \(U\). Define the shifted mollification of the localized function by

\[f_\varepsilon(y)=\int_{B_\varepsilon(0)}\eta_\varepsilon(z)f(y+\alpha\varepsilon e_n-z)\,dz.\]

The strict margin makes this function smooth on a neighbourhood of the smaller closed chart intersected with \(\overline U\). Differentiation and the weak derivative identity give the same formula with \(D_if\) in place of \(f\).

For the convergence estimate, extend the localized \(f\) and its interior weak derivatives separately by zero as \(L^p\) functions. Translation continuity and the mollifier estimate show that the displayed averages converge to \(f\) and \(D_if\) in \(L^p(U)\), respectively. This argument does not assume that the zero extension of \(f\) has the zero-extended weak derivative across \(\partial U\). The support remains away from the chart edges for small \(\varepsilon\), so the smoothed function can be set to zero outside the chart.

Compactness of \(\partial U\) gives finitely many such charts. Use a smooth partition \(\psi_0+\cdots+\psi_N=1\) on a neighbourhood of \(\overline U\), where \(\psi_0\) has compact support inside \(U\) and the other terms are supported strictly inside the boundary charts. Smooth \(\psi_0f\) by ordinary convolution and each \(\psi_if\) by the shifted construction, choosing each error less than \(\delta/(N+1)\). The finite sum is smooth near \(\overline U\) and has \(W^{1,p}\) error at most \(\delta\). If \(f\) is also continuous on \(\overline U\), these same approximations converge uniformly there: every averaging point stays in \(\overline U\) and approaches its centre, so uniform continuity applies.

Theorem 9. A measurable function belongs to \(W^{1,\infty}_{\rm loc}(U)\) if and only if its almost-everywhere equivalence class has a locally Lipschitz representative. Arbitrary changes on null sets need not preserve pointwise Lipschitz continuity.

Proof. Suppose first that \(f\in W^{1,\infty}_{\rm loc}(U)\). On a ball \(B\) whose closure is contained in a slightly larger ball \(B'\Subset U\), the mollifications satisfy

\[\|f^\varepsilon\|_{L^\infty(B)}\leq\|f\|_{L^\infty(B')},\qquad \|Df^\varepsilon\|_{L^\infty(B)}\leq\|Df\|_{L^\infty(B')}.\]

The fundamental theorem of calculus on line segments makes them uniformly Lipschitz on \(B\). A subsequence converges uniformly on every smaller closed ball by Arzelà–Ascoli, and its limit equals \(f\) almost everywhere by Theorem 5. These local continuous representatives agree on overlaps, since continuous functions that agree almost everywhere on an open set agree everywhere. They therefore define a locally Lipschitz representative on \(U\).

Conversely, on each sufficiently small ball let \(f\) be \(L\)-Lipschitz. Its restrictions to coordinate lines are Lipschitz, hence absolutely continuous, with derivatives bounded by \(L\) almost everywhere. Fubini and one-dimensional integration by parts give, for compactly supported smooth \(\varphi\),

\[\int f\partial_i\varphi=-\int(\partial_i f)\varphi,\qquad |\partial_i f|\leq L\quad\text{a.e.}\]

Here the partial derivatives agree almost everywhere with the classical derivative supplied by Rademacher’s theorem. Local boundedness of \(f\) and a finite covering of each compact subset prove \(f\in W^{1,\infty}_{\rm loc}(U)\).

Theorem 10. Let \(U\subset\mathbb R^n\) be bounded and open with Lipschitz boundary, and let \(1\leq p<\infty\). There is a unique bounded linear operator

\[T:W^{1,p}(U)\longrightarrow L^p(\partial U;\mathcal H^{n-1})\]

that agrees with restriction for \(f\in W^{1,p}(U)\cap C(\overline U)\). For every \(\varphi\in C^1(\mathbb R^n;\mathbb R^n)\),

\[\int_U f\operatorname{div}\varphi\,dx=-\int_U Df\cdot\varphi\,dx +\int_{\partial U}Tf\,\varphi\cdot\nu\,d\mathcal H^{n-1},\]

where \(\nu\) is the outward unit normal.

Definition 11. \(Tf\) is the trace of \(f\) on \(\partial U\), defined up to \(\mathcal H^{n-1}\)-null sets.

Proof of Theorem 10. First let \(f\in C^1(\overline U)\) be supported inside a boundary chart. On each vertical line, \(f\) vanishes near the top of the chart, so the fundamental theorem of calculus yields

\[|f(y',\gamma(y'))|\leq\int_{\gamma(y')}^{\text{top}}|\partial_nf(y',t)|\,dt.\]

The graph area element is \(\sqrt{1+|D\gamma|^2}\,dy'\leq\sqrt{1+L^2}\,dy'\). Integration proves the local \(L^1\) trace bound. Applying this to the finitely many localized pieces \(\psi_if\), and using \(D(\psi_if)=\psi_iDf+fD\psi_i\), gives

\[\int_{\partial U}|f|\,d\mathcal H^{n-1}\leq C_U\int_U(|Df|+|f|)\,dx.\]

Equivalently, absolute values can be smoothed by \(\beta_\varepsilon(t)=\sqrt{t^2+\varepsilon^2}-\varepsilon\), for which \(\beta_\varepsilon(0)=0\) and \(|\beta_\varepsilon'|\leq1\). This is the bound needed to pass to \(|f|\).

For \(p>1\), apply the \(L^1\) inequality to \(|f|^p\) and use the chain rule and Young’s inequality:

\[\int_{\partial U}|f|^p\,d\mathcal H^{n-1} \leq C_U\int_U\bigl(p|f|^{p-1}|Df|+|f|^p\bigr)\,dx \leq C_{U,p}\|f\|_{W^{1,p}(U)}^p.\]

Theorem 8 makes restriction extend uniquely to a bounded linear map \(T\). When \(f\) is continuous on \(\overline U\), the uniform approximation at the end of its proof shows that this extension equals its actual boundary values.

For smooth \(f\), integration by parts on a Lipschitz domain follows directly from graph coordinates. For a compactly supported smooth vector field \(F\) in a graph chart, vertical integration gives \(\int_U\partial_nF_n=-\int F_n(y',\gamma)\,dy'\). For \(i<n\), differentiating \(\int_{\gamma(y')}^{\text{top}}F_i(y',t)\,dt\) and integrating in \(y'\) gives \(\int_U\partial_iF_i=\int F_i(y',\gamma)\partial_i\gamma\,dy'\). Their sum is \(\int_{\partial U}F\cdot\nu\,d\mathcal H^{n-1}\); a partition of unity cancels the cutoff derivatives. Apply this identity to \(F=f\varphi\). Approximate a general \(f\) by Theorem 8. The two interior terms converge by \(L^p\) convergence on a bounded domain, and the boundary term converges by boundedness of \(T\) and finite surface measure. This proves the formula.

Theorem 12. Let \(U\) be bounded and open with Lipschitz boundary, \(1\leq p<\infty\), and \(\overline U\subset V\) for an open set \(V\). There is a bounded linear map

\[E:W^{1,p}(U)\longrightarrow W^{1,p}(\mathbb R^n)\]

such that \(Ef=f\) almost everywhere on \(U\). It can be chosen so that all \(Ef\) are supported in one fixed compact subset of \(V\).

Proof. In a graph chart flatten the boundary using

\[\Phi(y',t)=(y',\gamma(y')+t).\]

Both \(\Phi\) and its inverse are Lipschitz, and their Jacobians have absolute determinant one almost everywhere. Pullback by either map is bounded on \(W^{1,p}\) on the corresponding patches: for smooth functions the almost-everywhere chain rule, the area formula, and the bound on \(D\Phi\) prove this estimate; approximation from Theorem 6 and completeness pass the formula and estimate to Sobolev functions. The \(L^p\) limit is the intended composition because these bi-Lipschitz maps preserve null sets.

In flattened coordinates extend from \(t>0\) to \(t<0\) by even reflection, \(u(y',t)=u(y',-t)\). This gives a bounded extension across \(t=0\). To see that there is no additional measure on that plane, first use smooth approximations on the upper half-patch, as in Theorem 8. Their reflected functions are continuous and piecewise smooth; integration by parts on the two half-patches has cancelling boundary terms. Their weak derivatives are the reflected tangential derivatives and the sign-reversed normal derivative. The \(L^p\) norm of both function and gradient increases by at most a fixed factor. Passing to the \(W^{1,p}\) limit proves the same assertion for \(u\). Transport back by \(\Phi\); in original coordinates the reflection is \((x',x_n)\mapsto(x',2\gamma(x')-x_n)\).

Apply this construction to finitely many pieces of a fixed partition of unity near \(\partial U\), with supports away from chart edges; extend the interior piece by zero. Cutoffs inside the larger reflected patches allow each piece to be set to zero away from its chart. Their sum defines a bounded linear extension \(E_0\). Finally choose \(\chi\in C_c^\infty(V)\) with \(\chi=1\) near \(\overline U\) and set \(Ef=\chi E_0f\). The product rule gives the asserted norm bound and fixed compact support. All charts, cutoffs, and reflections are fixed independently of \(f\).

1. Sobolev Inequalities

Sobolev inequalities convert control of weak derivatives into stronger integrability, and in the supercritical range into continuity. For any \(1 \leq p < n\), the Sobolev conjugate is defined as

\[\begin{aligned} p^\star \triangleq \frac{np}{n-p} ~. \end{aligned}\]

Theorem 13. (Gagliardo–Nirenberg–Sobolev.) Let \(1\leq p<n\) and \(p^\star=np/(n-p)\). For every \(f\in W^{1,p}(\mathbb R^n)\),

\[\|f\|_{L^{p^\star}(\mathbb R^n)}\leq C(n,p)\|Df\|_{L^p(\mathbb R^n)}.\]

In particular, the inequality holds for \(f\in C_c^\infty(\mathbb R^n)\). The function-space hypothesis excludes nonzero constants on all of \(\mathbb R^n\).

Proof. Start with \(p=1\) and \(u\in C_c^1(\mathbb R^n)\). Integration along the \(i\)th coordinate gives

\[|u(x)|\leq F_i(\widehat x_i),\qquad F_i(\widehat x_i)=\int_{\mathbb R}|\partial_i u(x)|\,dx_i,\]

where \(\widehat x_i\) omits coordinate \(i\). Repeated Hölder inequalities give the integral estimate

\[\int_{\mathbb R^n}\prod_{i=1}^n F_i(\widehat x_i)^{1/(n-1)}\,dx \leq\prod_{i=1}^n\left(\int_{\mathbb R^{n-1}}F_i\right)^{1/(n-1)}.\]

For completeness, the \(n=2\) case is Fubini. For the induction step, integrate first in \(x_n\) and apply Hölder to the \(n-1\) factors with \(i<n\). Write \(G_i=\int F_i\,dx_n\). The remaining integral is bounded by

\[\left(\int F_n\right)^{1/(n-1)} \left(\int_{\mathbb R^{n-1}}\prod_{i<n}G_i^{1/(n-2)}\right)^{(n-2)/(n-1)},\]

by Hölder with exponents \(n-1\) and \((n-1)/(n-2)\). The induction hypothesis gives the desired product. Truncation and monotone convergence handle general nonnegative integrable factors.

Since \(|u|^{n/(n-1)}\leq\prod_iF_i^{1/(n-1)}\), it follows that

\[\|u\|_{n/(n-1)}\leq\prod_i\|\partial_i u\|_1^{1/n}\leq C_n\|Du\|_1.\]

For \(1<p<n\), apply this result to \(|u|^\alpha\), where \(\alpha=p(n-1)/(n-p)>1\). Hölder gives

\[\|u\|_{p^\star}^\alpha \leq C_n\alpha\int |u|^{\alpha-1}|Du| \leq C_n\alpha\|Du\|_p\|u\|_{p^\star}^{\alpha-1},\]

because \(\alpha n/(n-1)=p^\star\) and \((\alpha-1)p'=p^\star\). Divide unless \(u=0\), in which case the result is immediate.

Finally, cutoffs \(\chi_R\) with \(\chi_R=1\) on \(B_R\), \(|D\chi_R|\leq C/R\), followed by mollification approximate every \(f\in W^{1,p}(\mathbb R^n)\) by \(C_c^\infty\) functions in \(W^{1,p}\). The term \(fD\chi_R\) tends to zero in \(L^p\), while the remaining tails tend to zero by integrability. Applying the inequality to differences makes the approximants Cauchy in \(L^{p^\star}\). Their \(L^p\) limit identifies the \(L^{p^\star}\) limit with \(f\) almost everywhere.

Theorem 14. Let \(B=B_r(x_0)\), \(1\leq p<\infty\), \(f\in C^1(\overline B)\), and \(z\in\overline B\). Then

\[\int_B|f(y)-f(z)|^p\,dy \leq C(n,p)r^{n+p-1}\int_B|Df(y)|^p|y-z|^{1-n}\,dy.\]

This radial estimate itself does not require \(p<n\).

Proof. Along a ray \(y=z+\rho\omega\) inside the convex ball, the fundamental theorem of calculus and Hölder give

\[|f(z+\rho\omega)-f(z)|^p\leq\rho^{p-1}\int_0^\rho|Df(z+t\omega)|^p\,dt.\]

Let \(\ell(\omega)\leq2r\) be the length of that ray inside the ball (zero for outward rays at a boundary point). Multiply by \(\rho^{n-1}\), integrate \(0<\rho<\ell(\omega)\), and change the order of integration:

\[\int_0^{\ell(\omega)}\rho^{n+p-2}\int_0^\rho|Df(z+t\omega)|^p\,dt\,d\rho \leq\frac{(2r)^{n+p-1}}{n+p-1}\int_0^{\ell(\omega)}|Df(z+t\omega)|^p\,dt.\]

Integrating over \(\omega\) converts the last expression to the weighted integral in the statement. In dimension one, use the two ray directions; the weight is one.

Theorem 15. (Sobolev–Poincaré.) If \(1\leq p<n\), \(B=B_r(x_0)\), and \(f\in W^{1,p}(B)\), then, writing \([f]_{x_0,r}=f_B=\fint_Bf\),

\[\left(\fint_B|f-f_B|^{p^\star}\,dy\right)^{1/p^\star} \leq C(n,p)r\left(\fint_B|Df|^p\,dy\right)^{1/p}.\]

Proof. First integrate Theorem 14 in \(z\in B\). Since \(\int_B|y-z|^{1-n}\,dz\leq C_nr\) and Jensen’s inequality gives \(|f(y)-f_B|^p\leq\fint_B|f(y)-f(z)|^p\,dz\), we obtain the ordinary Poincaré inequality

\[\|f-f_B\|_{L^p(B)}\leq C(n,p)r\|Df\|_{L^p(B)}.\]

This argument starts with smooth \(f\); Theorem 8 on the ball passes it to \(W^{1,p}(B)\).

On the unit ball let \(h=f-f_B\) and extend \(h\) by Theorem 12. The extension bound, ordinary Poincaré inequality, and Theorem 13 give

\[\|h\|_{L^{p^\star}(B_1)}\leq\|Eh\|_{L^{p^\star}(\mathbb R^n)} \leq C\|D(Eh)\|_p\leq C\|h\|_{W^{1,p}(B_1)}\leq C\|Df\|_{L^p(B_1)}.\]

The change of variables \(y=x_0+rw\) gives the stated averaged inequality. Thus the subtraction of the mean is exactly what removes the extra \(L^p\) term in the extension estimate.

Theorem 16. (Morrey.) Let \(n<p<\infty\). Each \(f\in W^{1,p}(B_r(x_0))\) has a continuous representative on \(\overline{B_r(x_0)}\) satisfying, for all \(y,z\) there,

\[|f(y)-f(z)|\leq C(n,p)r\left(\fint_{B_r(x_0)}|Df|^p\,dw\right)^{1/p}.\]

If \(f\in W^{1,p}(\mathbb R^n)\), its representative is globally Hölder continuous with exponent \(1-n/p\), and

\[|f(y)-f(z)|\leq C(n,p)\|Df\|_{L^p(\mathbb R^n)}|y-z|^{1-n/p}.\]

Moreover \(f^\star(x)=\lim_{r\downarrow0}[f]_{x,r}\) exists at every \(x\) and equals this representative.

\[|f(y)-f(z)|\leq C_n\int_B|Df(w)|\bigl(|w-y|^{1-n}+|w-z|^{1-n}\bigr)\,dw.\]

For \(p'=p/(p-1)\), the condition \(p>n\) is exactly \((n-1)p'<n\). Consequently

\[\left(\int_B|w-y|^{(1-n)p'}\,dw\right)^{1/p'} \leq C(n,p)r^{1-n/p},\]

uniformly for \(y\in\overline B\). Hölder proves the first estimate for smooth functions.

Choose smooth approximations \(f_j\to f\) in \(W^{1,p}(B)\) by Theorem 8. Apply the estimate to \(f_j-f_k\) and bound its mean by \(|B|^{-1/p}\|f_j-f_k\|_p\). This gives

\[\|f_j-f_k\|_{L^\infty(B)}\leq |B|^{-1/p}\|f_j-f_k\|_p +C(n,p)r^{1-n/p}\|D(f_j-f_k)\|_p\longrightarrow0.\]

The uniform limit on \(\overline B\) is continuous, equals \(f\) almost everywhere, and satisfies the same oscillation bound. Representatives on overlapping balls agree by continuity. For \(f\) on \(\mathbb R^n\) and \(y\ne z\), choose a ball centred at their midpoint with radius \(|y-z|\). The first estimate then gives the global Hölder bound. The averages of a continuous function tend to its value at every point, proving the assertion about \(f^\star\).

Theorem 17. (Rellich compactness.) Suppose \(U\subset\mathbb R^n\) is bounded and open with Lipschitz boundary and \(1<p<n\). Every bounded sequence \((f_k)\) in \(W^{1,p}(U)\) has a subsequence converging to some \(f\in W^{1,p}(U)\) in \(L^q(U)\) for every \(1\leq q<p^\star\).

Proof. Use the fixed extension from Theorem 12 and put \(F_k=Ef_k\). Their \(W^{1,p}(\mathbb R^n)\) norms are bounded by a constant \(M\), and their supports lie in a single compact set \(K\).

First, for \(F\in W^{1,p}(\mathbb R^n)\),

\[\|F(\cdot-h)-F\|_p\leq |h|\|DF\|_p.\]

For smooth \(F\) this follows from \(F(x-h)-F(x)=-\int_0^1DF(x-th)\cdot h\,dt\) and Minkowski; smooth approximation proves the general case. Averaging with the mollifier gives the uniform estimate

\[\sup_k\|F_k^\varepsilon-F_k\|_p\leq C M\varepsilon.\]

Second, Hölder applied to convolution gives, for each fixed \(\varepsilon>0\),

\[\|F_k^\varepsilon\|_\infty\leq\|\eta_\varepsilon\|_{p'}M =C\varepsilon^{-n/p}M,\qquad \|DF_k^\varepsilon\|_\infty\leq\|D\eta_\varepsilon\|_{p'}M =C\varepsilon^{-1-n/p}M.\]

Their supports lie in \(K+\overline{B_\varepsilon}\). Thus Arzelà–Ascoli gives a subsequence converging uniformly, hence in \(L^p\), for each fixed \(\varepsilon\).

Take nested subsequences for \(\varepsilon_\ell\downarrow0\) and then their diagonal. To see this diagonal is Cauchy in \(L^p\), fix \(\ell\) and use

\[\|F_{k_i}-F_{k_j}\|_p\leq \|F_{k_i}-F_{k_i}^{\varepsilon_\ell}\|_p+ \|F_{k_i}^{\varepsilon_\ell}-F_{k_j}^{\varepsilon_\ell}\|_p+ \|F_{k_j}^{\varepsilon_\ell}-F_{k_j}\|_p.\]

Choose \(\ell\) to make the first and third terms small uniformly, then \(i,j\) to control the middle term. In particular, \(f_{k_j}\to f\) in \(L^p(U)\).

Because \(p>1\), boundedness of the gradients in the reflexive space \(L^p(U;\mathbb R^n)\) gives a further subsequence with \(Df_{k_j}\rightharpoonup G\). For every test function,

\[\int_U f\partial_i\varphi=\lim_j\int_U f_{k_j}\partial_i\varphi =-\lim_j\int_U\partial_if_{k_j}\varphi=-\int_U G_i\varphi.\]

Hence \(Df=G\) and \(f\in W^{1,p}(U)\); this is the step that uses reflexivity.

Theorem 13 and the extension bound give a uniform \(L^{p^\star}\) bound, and Fatou gives the same bound for \(f\). For \(p<q<p^\star\), choose \(0<\theta<1\) with \(1/q=\theta/p+(1-\theta)/p^\star\). Then

\[\|f_{k_j}-f\|_q\leq\|f_{k_j}-f\|_p^\theta \|f_{k_j}-f\|_{p^\star}^{1-\theta}\longrightarrow0.\]

For \(1\leq q\leq p\), use \(\|g\|_q\leq |U|^{1/q-1/p}\|g\|_p\). The same subsequence works for all these exponents.

The exclusion of the endpoint is substantive. Choose \(x_0\in U\) and nonzero \(\phi\in C_c^\infty(B_1)\), and for sufficiently small \(\varepsilon\) set

\[f_\varepsilon(x)=\varepsilon^{1-n/p}\phi((x-x_0)/\varepsilon).\]

Its gradient norm and \(L^{p^\star}\) norm are fixed, while \(\|f_\varepsilon\|_p=\varepsilon\|\phi\|_p\) and \(f_\varepsilon\to0\) almost everywhere. It cannot have a strongly convergent subsequence in \(L^{p^\star}\). For \(p=1<n\), the same mollification proof gives compactness in \(L^q\), \(q<n/(n-1)\), but the limit is only guaranteed to be in \(BV(U)\): testing the variation definition below and passing to the \(L^1\) limit gives a bounded distributional derivative. Smooth approximations to the indicator of a ball compactly contained in \(U\) show that this limit need not lie in \(W^{1,1}(U)\).

2. Functions of Bounded Variation

Functions whose weak first partial derivatives are Radon measures are called functions of bounded variation on \(\mathbb R^n\). This permits singular derivatives, including the boundary measure arising from the characteristic function of a smooth domain. It is not obvious that any of the usual rules of calculus apply to functions whose first derivatives are merely measures, so it is of interest to define and study functions of bounded variations. Henceforth, let \(U \subset \mathbb R^n\) be an open set, unless otherwise stated.

Definition 18. An \(f\in L^1(U)\) belongs to \(BV(U)\) if

\[|Df|(U):=\sup_{\varphi\in C_c^1(U;\mathbb R^n),\ |\varphi|\leq1} \int_U f\operatorname{div}\varphi\,dx<\infty.\]

The local space \(BV_{\rm loc}(U)\) consists of \(f\in L^1_{\rm loc}(U)\) for which the analogous supremum is finite on every \(V\Subset U\).

Definition 19. A Lebesgue measurable set \(E\subset\mathbb R^n\) has finite perimeter in \(U\) if

\[P(E;U):=\sup_{\varphi\in C_c^1(U;\mathbb R^n),\ |\varphi|\leq1} \int_{E\cap U}\operatorname{div}\varphi\,dx<\infty.\]

This requires a finite distributional derivative, not a finite volume. If also \(|E\cap U|<\infty\) (in particular if \(U\) is bounded), this is equivalent to \(\mathbb1_E\in BV(U)\). Locally finite perimeter means \(\mathbb1_E\in BV_{\rm loc}(U)\).

The following theorem explains the notation \(|Df|\): the distributional derivative is a vector-valued Radon measure, and \(|Df|\) is its total variation measure.

Theorem 20 (\(BV_{\rm loc}(U)\) Structure Theorem). Suppose \(f \in BV_{\rm loc}(U)\). Then, there exists a Radon measure \(\mu\) on \(U\), and a \(\mu\)-measurable function \(\sigma : U \to \mathbb R^n\) such that the following conditions hold:

\(|\sigma(x)|=1\) for \(\mu\)-a.e.;
for all \(\varphi \in \mathscr C^1_c (U; \mathbb R^n)\),

\[\begin{aligned} \int_U f \operatorname{div}\varphi \dif x = - \int_U \varphi \cdot \sigma \dif \mu ~. \end{aligned}\]

Proof. The trick is to define a linear functional

\[\begin{aligned} L : \mathscr C^1_c (U; \mathbb R^n) \to \mathbb R : \varphi \mapsto -\int_U f \operatorname{div}\varphi \dif x ~. \end{aligned}\]

This is linear by the definition of an integral. Since \(f \in BV_{\rm loc}(U)\),

\[\begin{aligned} \alpha(V) \triangleq \sup_{\varphi \in \mathscr C^1_c (V;\mathbb R^n) : |\varphi|\leq 1} \int_V f \operatorname{div} \varphi \dif x < \infty ~. \end{aligned}\]

Therefore, \(|L(\varphi)|\leq \alpha(V) \|\varphi\|_{L^\infty}\), hence giving us a Lipschitz constant-like bound. It remains to verify that we can apply the Riesz representation theorem.

Pick any compact set \(K \subset U\), and choose an open set \(V\) such that \(K \subset V \subset_c U\). For every \(\varphi \in \mathscr C_c(U; \mathbb R^n)\) such that \(\operatorname{spt}(\varphi) \subset K\), we can pick some sequence of approximating functions \(\varphi_k \in \mathscr C^1_c (V; \mathbb R^n)\) such that \(\varphi_k \to \varphi\) uniformly on \(V\). The limit

\[\begin{aligned} L^*(\varphi) \triangleq \lim\limits_{k\to\infty} L(\varphi_k) \end{aligned}\]

exists by the estimate on \(|L(\varphi)|\), and this is independent of the choice of the sequence of \(\varphi_k\). Therefore, there exists a unique extension of \(L\) to a linear functional \(L^* : \mathscr C_c (U; \mathbb R^n) \to \mathbb R\) such that \(\sup \{L(\varphi) : \varphi \in \mathscr C_c(U; \mathbb R^n) ~,~ \operatorname{spt}(\varphi) \subset K ~,~ |\varphi| \leq 1\} < \infty\) for each compact set \(K \subset U\). Applying the Riesz representation theorem to the components gives a unique vector-valued Radon measure representing this functional. The remaining step is its polar decomposition, detailed below.

To finish Theorem 20, denote the representing vector measure by \(Df=(D_1f,\ldots,D_nf)\) and its total variation by \(\mu=|Df|\). Each component is absolutely continuous with respect to \(\mu\). The Radon–Nikodym theorem therefore gives \(Df=\sigma\mu\). The variation of a vector measure with density \(\sigma\) is \(|\sigma|\mu\) (first check simple densities on disjoint measurable sets, then approximate in \(L^1(\mu)\) on compact sets). Since \(\mu\) is already the variation of \(Df\), \(|\sigma|=1\) \(\mu\)-almost everywhere. Thus

\[\int_U f\operatorname{div}\varphi\,dx=-\int_U\varphi\cdot\sigma\,d\mu.\]

The vector measure is unique by uniqueness in Riesz representation; hence \(\mu\) is unique and \(\sigma\) is unique \(\mu\)-almost everywhere. The dual characterization of total variation by continuous compactly supported fields of norm at most one, followed by uniform smooth approximation with arbitrarily small rescaling, identifies \(\mu(V)\) with the supremum in Definition 18 for each open \(V\subset U\). In the global case its total mass is finite.

Example 21. (Sobolev functions.) If \(f\in W^{1,1}_{\rm loc}(U)\), then for \(V\Subset U\) and \(\varphi\in C_c^1(V;\mathbb R^n)\) with \(|\varphi|\leq1\),

\[\int_Uf\operatorname{div}\varphi\,dx=-\int_UDf\cdot\varphi\,dx\leq\int_V|Df|\,dx<\infty.\]

Here the Sobolev gradient is a function; its associated derivative measure is \(Df\,dx\), with total variation \(|Df|\,dx\). Its polar density is \(\sigma=Df/|Df|\) where \(Df\ne0\), and can be set to zero on the set where \(Df=0\) (which has zero variation measure). Consequently

\[W^{1,1}_{\rm loc}(U)\subset BV_{\rm loc}(U),\qquad W^{1,1}(U)\subset BV(U).\]

Hölder on relatively compact subsets also gives \(W^{1,p}_{\rm loc}(U)\subset BV_{\rm loc}(U)\) for every \(1\leq p\leq\infty\).

Example 22. (Smooth sets and singular derivatives.) Let \(E\) be a smooth open set with \(\mathcal H^{n-1}(\partial E\cap K)<\infty\) for every compact \(K\subset U\). Gauss–Green gives, for \(\varphi\in C_c^1(U;\mathbb R^n)\),

\[\int_U\mathbb1_E\operatorname{div}\varphi\,dx =\int_{\partial E\cap U}\varphi\cdot\nu\,d\mathcal H^{n-1},\]

where \(\nu\) is the outward normal. In particular, when \(|\varphi|\leq1\) and its support lies in \(V\Subset U\), this integral is at most \(\mathcal H^{n-1}(\partial E\cap\overline V)<\infty\). Hence \(\mathbb1_E\in BV_{\rm loc}(U)\) and uniqueness in Theorem 20 identifies

\[D\mathbb1_E=-\nu\,\mathcal H^{n-1}\mathbin{\restriction}(\partial E\cap U),\qquad |D\mathbb1_E|=\mathcal H^{n-1}\mathbin{\restriction}(\partial E\cap U).\]

The polar vector in Theorem 20 is \(\sigma=-\nu\). If the measure-theoretic outward normal is written \(\nu_E\), its convention is \(\nu_E=-\sigma=\nu\). The perimeter measure, sometimes denoted \(\|\partial E\|\), is \(|D\mathbb1_E|\).

If \(\partial E\cap U\) is nonempty, its smooth hypersurface measure is locally positive but singular with respect to \(n\)-dimensional Lebesgue measure. Therefore \(D\mathbb1_E\) cannot be represented by a locally integrable function, so \(\mathbb1_E\notin W^{1,1}_{\rm loc}(U)\). This conclusion needs a boundary inside \(U\); it is false, for example, if \(E=\mathbb R^n\). For any nonempty open \(U\), choosing a ball with closure contained in \(U\) supplies an indicator with finite volume and perimeter, proving

\[W^{1,1}_{\rm loc}(U)\subsetneqq BV_{\rm loc}(U),\qquad W^{1,1}(U)\subsetneqq BV(U).\]

Reading: Carmo (1992), Riemannian geometry.